Hilbert's Paradox of the Grand Hotel

Hilbert's Paradox of the Grand Hotel is a thought experiment created by the mathematician David Hilbert. It is meant to illustrate some of the counterintuitive properties of infinite sets. Let's break it down using a simple example.

Imagine a hotel with a finite number of rooms. Suppose all the rooms are occupied, and a new guest arrives. Since there are no vacancies, the new guest cannot be accommodated, and the hotel is considered full.

Now, imagine a hotel with an infinite number of rooms, called the Grand Hotel. Every room is occupied by a guest. At first glance, you might think that the Grand Hotel is also full, and that no more guests can be accommodated. However, this is where the paradox comes into play.

If a new guest arrives at the Grand Hotel, the manager can simply ask every current guest to move to the room with the room number one higher than their current room number (i.e., the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on). By doing this, the first room becomes vacant, and the new guest can be accommodated.

The paradox becomes even more counterintuitive when you consider different scenarios:

1. An infinite number of new guests ...